Explanation

(a) For hydrogen or hydrogen type atoms
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{\lambda}$= ${RZ}^{2}\left( \frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}} \right)$
In the transition from $ni \rightarrow \ nf$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \lambda\ \propto \ \frac{1}{Z^{2}\left( \frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}} \right)}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\lambda_{2}}{\lambda_{1}} = \ \frac{Z_{1}^{2}\left( \frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}} \right)_{1}}{Z_{2}^{2}\left( \frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}} \right)_{2}}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda_{2} = \ \frac{\lambda_{1}Z_{1}^{2}\left( \frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}} \right)_{1}}{Z_{2}^{2}\left( \frac{1}{n_{f}^{2}} - \frac{1}{n_{i}^{2}} \right)_{2}}$
Substituting the values, we have
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{(6561)(1)^{2}\left( \frac{1}{2^{2}} - \frac{1}{3^{2}} \right)}{(2)^{2}\left( \frac{1}{2^{2}}\ - \ \frac{1}{4^{2}} \right)}$ $= \ 1215\ \mathring{\mathrm{A}}$